A year ago, I really wasn't satisfied with this part of the project. I thought there was a lot more potential here than was actually realized in class, both in terms of getting students to talk about their work, and in getting students to gain practice in identifying errors and suggesting corrections.

I am much happier with the results of this version of the project. First of all, on Part 4a, I was surprised by how useful it was for kids to simply quote each other's work. I asked them to "Give an example of a sentence where your partner uses one of the inverse properties," which led to great discussions and gave students another chance to look closely at the details of one solution. It got me thinking that maybe citation is a prerequisite to critique, and that perhaps a reason my students have struggled with critique in the past is that they need more help looking through the work and figuring out what to critique in the first place. I was equally excited about how effective it was to have students interview each other about the work. Having great conversations about a math concept or problem is a skill, and this activity scaffolded toward mastery of it.

When students reflected on the project before submitting it, many noted how useful this part of the project was. Look at this student's reflection, for example: he "learned to professionally criticize my partner without being rude," and "to not be as mad or upset at my partner" when receiving feedback. Sure we're algebra teachers - but I'll take lessons like these over rote skills any day!

Part 4b was just as engaging, and I definitely plan on using a structure like this more often in the future. Here is a spreadsheet of student responses. The unexpected benefit here was that I got learn a little more about student dispositions toward math. The "teacher voice" that some students take on while giving feedback to these fictional students can tell a lot about how they feel about math, making mistakes, and receiving feedback. It's also useful to see when a student "honestly can't see what's wrong" with a solution. That gives me the opportunity to remediate immediately, helping students to fill specific gaps in their knowledge.

Today's opener (it's also on the first three slides of today's Prezi) is a repeat of one of the problems from the review problem set, "What Can You Do So Far?" This problem set was the focus of yesterday's class. Over the next two weeks, the problem set will serve to motivate review of a variety of different problems, and I'll reference it consistently with students as the unit comes to a close.

For today's opener we think about extending the Addition and Multiplication tables from the Number Line Project out to larger numbers. Thinking about the set of locations at which a given number can be found on the Addition and Multiplication tables paves the way for students to think about what the graphs

x + y = n

xy = n

will look like.

Thanks to our experience in the field, you or I might recognize the shapes of these two functions. But for students I'm trying to build some background knowledge before delving into that later in the year. For now, I want students to understand why a given number on the addition table will always have the same neighbors, while a number on the multiplication table will find itself in different company depending on its location.

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Resources

Today's Prezi slides continue by giving students an overview of the work that they'll engage in during today's and tomorrow's classes. Students will critique each other's work on Part 2 of the Linear Equation Project. They will then revise their work in some way, before submitting the project two days from now.

I want critique to become something that's not just normal, but indispensable in this class, and that work begins here. It's hard work. It requires a great deal of attention from students, and a great deal of patience of flexibility from the teacher.

To begin our work, I distribute Part 3 of the Linear Equation Project, which is called "Critique and Final Draft". Students will find a partner, critique that partner's work, then produce a final draft of the equation and justification that they created on Part 2 of the project. There are four areas that I've laid out for students to examine. I introduce these by navigating around the Part 3 handout on slides 7 through 12 of the Prezi, and by briefly discussing the meaning of each.

For your perusal, I've included an example of critique from an average-level student: page 1, and page 2. I would like for this student to go into greater depth here, but I also think that her level of engagement is enough for her to learn and improve over the course of the year. Additionally, here are before and after critique snapshots of another student's work.

As this work continues, there is a lot that can happen. Students will have a chance to solidify their knowledge of the arithmetic properties, which they will continue to gather on the note catcher.

With a little less than 10 minutes left in the class, I distribute cover sheets for the Linear Equation Project. Everyone still has more to do; most students need to finish their critique, and everyone will need time to complete their final drafts, and I will provide them with that time tomorrow. I don't want to rush anyone, but I want them to have the rubric (which is on the back of the cover sheet) and I want everyone to understand what I'll be collecting two days from now.

There are three things to note about the cover sheet:

The submission checklist tells students exactly what to hand in. I'll have to direct their attention to this a few times over the next few days. Having it in writing helps me be clear and consistent, as students will continually ask me to repeat my expectations for what gets handed in.

Four reflection questions are at the bottom of the cover sheet. I always ask my students to reflect in some way when they submit a project. There are different levels of rigor that students apply to their responses here, but all are instructive to me as a teacher, and I believe that for students, the act of saying what they've learned is always productive. Take a look at some of the samples of student responses I've included here and think about what you're learning about each of these students.

The rubric is on the back. An important growth area in my practice is that I'm learning to write rubrics that are simple for students to understand, quick and user-friendly for me, and that offer plenty of space for me to give useful feedback to kids. I feel like I'm moving in that direction with this rubric, but in doing so, I've kind of left behind the detailed description of each mastery-level. This is a work in progress for me, and I welcome your feedback.

Wow - I really appreciate the way you put that, Aaron. It's important to note that the transition away from teacher authority can be difficult for kids to make! Some of my brightest students have a hard time with the way I'm always asking what they think about a problem, and they say they wish I'd just "tell them the answer" more often. A common line among my students is that I "always answer their questions with more questions." What's gratifying to witness is that over the course of the year, that sentiment goes from being a complaint to being a reason they love the class. By the Spring, students have the ability to know when they're right, and to be each other's best teachers.

Hi James - In critique exemplar 3 the student says that they learned different people explain things differently according to how they understand it. I love that, and it completely justifies the way in which you ask your students to work. I hope that by what we do in our classrooms all students come to see that and come to value it. Students sometimes get the mis-guided idea that mathematical authority comes from a book or a source like the teacher. The more we can help them see the mathematical authority coming from sound student reasoning, the better off they will be.